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Karnataka Class 12 Commerce Maths Determinants Complete Details

Karnataka Class 12 Commerce Maths Determinants Complete Details

Karnataka Class 12 Commerce Maths Determinants : Here our team members provides you Karnataka Class 12 Commerce Maths Determinants Complete  Notes in pdf format. Here we gave direct links for you easy to download Karnataka Class 12 Commerce Maths Determinants Complete Notes. Karnataka Class 12 Commerce Maths Determinants topics are Applications in finding the area under simple curves, especially lines, circles/parabolas/ellipses (in standard form only), Area between any of the two above said curves (the region should be clearly identifiable). Download these Karnataka Class 12 Commerce Maths Determinants Complete Notes and read well.

Download here Karnataka Class 12 Commerce Maths Complete Notes In PDF Format 

Karnataka Class 12 Commerce Maths Determinants Complete Details

Karnataka Class 12 Commerce Maths Determinants : This article is about determinants in mathematics. For determinants in epidemiology, see risk factor. In linear algebra, the determinant is a useful value that can be computed from the elements of a square matrix. The determinant of a matrix A is denoted det(A), det A, or | A |. It can be viewed as the scaling factor of the transformation described by the matrix.

In the case of a 2 × 2 matrix, the specific formula for the determinant is

{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b\\c&d\end{vmatrix}}=ad-bc.\end{aligned}}}

Similarly, suppose we have a 3 × 3 matrix A, and we want the specific formula for its determinant | A |:

{\displaystyle {\begin{aligned}|A|={\begin{vmatrix}a&b&c\\d&e&f\\g&h&i\end{vmatrix}}&=a\,{\begin{vmatrix}e&f\\h&i\end{vmatrix}}-b\,{\begin{vmatrix}d&f\\g&i\end{vmatrix}}+c\,{\begin{vmatrix}d&e\\g&h\end{vmatrix}}\\&=aei+bfg+cdh-ceg-bdi-afh.\end{aligned}}}

Each determinant of a 2 × 2 matrix in this equation is called a “minor” of the matrix A. The same sort of procedure can be used to find the determinant of a 4 × 4 matrix, the determinant of a 5 × 5 matrix, and so forth.

Determinants occur throughout mathematics. For example, a matrix is often used to represent the coefficients in a system of linear equations, and the determinant can be used to solve those equations, although more efficient techniques are actually used, some of which are determinant-revealing and consist of computationally effective ways of computing the determinant itself. The use of determinants in calculus includes the Jacobin determinant in the change of variables rule for integrals of functions of several variables. Determinants are also used to define the characteristic polynomial of a matrix, which is essential for eigenvalue problems in linear algebra. In analytic geometry, determinants express the signed n-dimensional volumes of n-dimensional paralleled. Sometimes, determinants are used merely as a compact notation for expressions that would otherwise be unwieldy to write down.

Karnataka Class 12 Commerce Maths Determinants Complete Details

Karnataka Class 12 Commerce Maths Determinants : For our definition of determinants, we express the determinant of a square matrix A in terms of its cofactor expansion along the first column of the matrix. This is different than the definition in the textbook by Leon: Leon uses the cofactor expansion along the first row. It will take some work, but we shall later see that this is equivalent to our definition.

Formally, we define the determinant as follows:

Definition Let A be a n×n matrix. Then the determinant of A is defined by the following. If A is 1 × 1 so that A = (a1,1), then det(A) = a1,1 . Otherwise, if n > 1,

In this section, we will study properties determinants have and we will see how these properties can help in computing the determinant of a matrix. We will also see how these properties can give us information about matrices.

Calculating the Determinant

First of all the matrix must be square (i.e. have the same number of rows as columns). Then it is just basic arithmetic. Here is how:

For a 2×2 Matrix

For a 2×2 matrix (2 rows and 2 columns):

A Matrix

The determinant is:

|A| = ad − bc
“The determinant of A equals a times d minus b times c”

It is easy to remember when you think of a cross:

  • Blue is positive (+ad),
  • Red is negative (−bc)
A Matrix

Example:

A Matrix

|B|= 4×8 − 6×3
= 32−18
= 14

For a 3×3 Matrix

For a 3×3 matrix (3 rows and 3 columns):

A Matrix

The determinant is:

|A| = a(ei − fh) − b(di − fg) + c(dh − eg)
“The determinant of A equals … etc”

It may look complicated, but there is a pattern:

A Matrix

To work out the determinant of a 3×3 matrix:

  • Multiply a by the determinant of the 2×2 matrix that is not in a‘s row or column.
  • Likewise for b, and for c
  • Add them up, but remember that b has a negative sign!

As a formula (remember the vertical bars || mean “determinant of”):

A Matrix
“The determinant of A equals a times the determinant of … etc”

Example:

A Matrix

|C|= 6×(−2×7 − 5×8) − 1×(4×7 − 5×2) + 1×(4×8 − (−2×2))
= 6×(−54) − 1×(18) + 1×(36)
= −306

For 4×4 Matrices and Higher

The pattern continues for 4×4 matrices:

  • plus a times the determinant of the matrix that is not in a‘s row or column,
  • minus b times the determinant of the matrix that is not in b‘s row or column,
  • plus c times the determinant of the matrix that is not in c‘s row or column,
  • minus d times the determinant of the matrix that is not in d‘s row or column,

A Matrix

As a formula:

A Matrix

Notice the +−+− pattern (+a… b… +c… d…). This is important to remember.

The pattern continues for 5×5 matrices and higher. Usually best to use a Matrix Calculator for those!

Karnataka Class 12 Commerce Maths Determinants Complete Details

Karnataka Class 12 Commerce Maths Determinants : Here is the theorem. Theorem. Let A be an n by n matrix. Then the following conditions hold.

  1. If we multiply a row (column) of A by a number, the determinant of A will be multiplied by the same number.
  2. If the i-th row (column) in A is a sum of the i-th row (column) of a matrix B and the i-th row (column) of a matrix C and all other rows in B and C are equal to the corresponding rows in A (that is B and C differ from A by one row only), then det(A)=det(B)+det(C).
  3. If two rows (columns) in A are equal then det(A)=0.
  4. If we add a row (column) of A multiplied by a scalar k to another row (column) of A, then the determinant will not change.
  5. If we swap two rows (columns) in A, the determinant will change its sign.
  6. det(A)=det(AT).

Proof. 1. In the expression of the determinant of A every product contains exactly one entry from each row and exactly one entry from each column. Thus if we multiply a row (column) by a number, say, k, each term in the expression of the determinant of the resulting matrix will be equal to the corresponding term in det(A) multiplied by k. Therefore the determinant of the resulting matrix will be equalk*det(A).

2. We have that A(i,j)=B(i,j)+C(i,j) for every j=1,2,…,n. Consider the expression for det(A). Each term in this expression contains exactly one factor from the i-th row of A. Conside one of these terms:

(-1) p A(p1,1) A(p2,2)…A(i,j)… A(pn,n)

(A(i,j) is the factor from i-th row in this product).

Replace A(i,j) by B(i,j)+C(i,j):

(-1) p A(p1,1) A(p2,2)…(B(i,j)+C(i,j))… A(pn,n)

Multiply through:

(-1) p A(p1,1) A(p2,2)…B(i,j)… A(pn,n)+
(-1) p A(p1,1) A(p2,2)…C(i,j)… A(pn,n)

If we add only the terms containing B‘s, we get the determinant of B; if we add all terms containing C‘s, we get the determinant of C. Thus det(A)=det(B)+det(C).


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3. Suppose that the i-th row in A is equal to the j-th row of A, that is A(i,k)=A(j,k) for every k=1,2,…,n. Consider an arbitrary product in the expression of det(A):

A(p1,1) A(p2,2)…A(i,k1)…A(j,k2)… A(pn,n)

(we use the fact that this product contains one factor from the i-th row and one factor from the j-th row, and we assume that i occurs before j; the case when i occurs further than j is similar). Consider also the product corresponding to the permutation p‘ obtained from p by switching i and j:

A(p1,1) A(p2,2)…A(j,k1)…A(i,k2)… A(pn,n)

Now since

A(i,k1)=A(j,k1), A(j,k2)=A(i,k2)

these terms are equal. But they occur in the expression of det(A) with opposite signs (remember that p‘ is obtained from p by one transposition). Thus these products kill each other in det(A). Therefore each term in det(A) gets killed when we combine like terms in det(A), so det(A)=0.

4. Let matrix B be obtained from matrix A by adding the j-th row multiplied by k to the i-th row. Let us represent A as a column of rows:

                     [ r1 ]
                       ...
                     [ ri ]
                       ...
                     [ rj ]
                       ...
                     [ rn ]

Then B has the following form:

                     [ r1                ]
                       ...
                     [ ri+krj ]
                       ... 
                     [ rj                ]
                       ...
                     [ rn                ]

By property 2 we can conclude that det(B) is equal to the sum of determinants of two matrices:

                     [ r1 ]
                       ...
                     [ ri ]
                       ...
                     [ rj ]
                       ...
                     [ rn ]

and

                     [ r1  ]
                       ...
                     [ krj ]
                       ...
                     [ rj  ]
                       ...
                     [ rn  ]

The first of these matrices is A. Let us denote the second one by C. So det(B)=det(A)+det(C). By property 1 det(C) is k times the determinant of the following matrix:

                     [ r1 ]
                       ...
                     [ rj ]
                       ...
                     [ rj ]
                       ...
                     [ rn ]

But this matrix has two equal rows, therefore its determinant is equal to 0 (property 3). Thus det(C)=0 and det(B)=det(A). The proof is complete.

5. Suppose that we swap the i-th row and the j-th row of matrix A. Represent A as a column of rows:

                     [ r1 ]
                       ...
                     [ ri ]
                       ...
                     [ rj ]
                       ...
                     [ rn ]

In order to swap ri and rj we can do the following procedure:

  1. Add the j-th row to the i-th row:
                         [ r1               ]
                           ...
                         [ ri+rj ]
                          ...
                         [ rj               ]
                           ...
                         [ rn               ]
    
  2. Subtract the i-th row of the resulting matrix from the j-th row:
                         [ r1               ]
                           ...
                         [ ri+rj ]
                           ...
                         [ -ri              ]
                           ...
                         [ rn               ]
    
  3. Add the j-th row of the resulting matrix to the i-th row:
                         [ r1  ]
                           ...
                         [ rj  ]
                           ...
                         [ -ri ]
                           ...
                         [ rn  ]
    
  4. Multiply the j-th row of the resulting matrix by -1:
                         [ r1 ]
                           ...
                         [ rj ]
                           ...
                         [ ri ]
                           ...
                         [ rn ]
    

     

     

By the properties that we already proved all operations of this procedure except the very last one do not change the determinant. The last operation changes the sign of the determinant. The proof is complete

Download here Karnataka Class 12 Commerce Maths Determinants Complete Details In PDF Format 

Karnataka Class 12 Commerce Maths Determinants Complete Details

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